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21st-octave temperaments: Difference between revisions

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This page collectes temperaments with a period of 1/21 of an octave.
{{Infobox fractional-octave|21}}
This page collects temperaments with a period of 1/21 of an [[octave]].


Although 21edo itself is not remarkably accurate for low-complexity harmonics, some temperaments which are multiples of 21, such as {{EDOs|441, 1407, and 1848}} are. 441 and 1848 are also members of [[zeta]] edo list.
Although [[21edo]] itself is not remarkably accurate for low-complexity harmonics, some temperaments which are multiples of 21, such as {{EDOs|441, 1407, and 1848}} are. 441 and 1848 are also members of the [[zeta]] edo list.


== Akjayland ==
Temperaments discussed elsewhere include
{{See also| Landscape microtemperaments #Akjayland }}
* ''[[Akjayland]]'' → [[Landscape microtemperaments #Akjayland|Landscape microtemperaments]]


[[Subgroup]]: 2.3.5.7
== 21-23-commatic ==
Subgroup: 2.23


[[Comma list]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}
Comma list: {{monzo|95 0 0 0 0 0 0 0 -21}}


[[Mapping]]: [{{val| 21 1 38 102 }}, {{val| 0 3 1 -4 }}]
{{Mapping|legend=2|21 95}}


Mapping generators: ~1323/1280, ~131072/91875
: Mapping generator: ~529/512 = 1\21


[[Optimal tuning]] ([[CTE]]): ~131072/91875 = 614.9354
[[Support]]ing [[ET]]s: 21N, N = 1 to 96, largest: [[2016edo]]


[[Optimal GPV sequence]]: {{val list| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
== Scandium ==
Described as the 525 & 1911 temperament, and named after the 21st element for splitting the octave into 21 parts. Coincidentally, ''Encyclopaedia Britannica'' entry for scandium was written in the year 1911 which was used as the reason for the naming. Remarkably, unlike akjayland or many temperaments in the thousands which contain 3edo as a subset, it is ''not'' a landscape system. [[39/32]] is mapped into 6\21 and [[23/16]] is, as usual, mapped into 11\21.


[[Badness]]: 0.0309
Subgroup: 2.3.5.7


=== Vasca ===
Comma list: {{monzo|47 -7 -7 -7}}, {{monzo|-29 0 27 -12}}
Vasca can be described as the 357 & 525 temperament, extended as high as the 23-limit. It tempers out the {{monzo| 95 0 0 0 0 0 0 0 -21 }}, and sets a stack of 21 [[23/16]]'s equal with 11 octaves. The name derives from elements vanadium (23) and scandium (21), since this uses the 23rd harmonic, which itself is extremely well represented in 21edo.


Subgroup: 2.3.5.7.11
{{Mapping|legend=1| 21 0 59 82 | 0 13 -4 -9 }}


Comma list: 3025/3024, 102487/102400, {{monzo| 39 -4 -11 -5 2 }}
: Mapping generators: ~403368/390625 = 1\21, ~160/147


Mapping: [{{val| 21 4 39 98 58 }}, {{val| 0 6 2 -8 3 3 }}]
[[Optimal tuning]] ([[CTE]]): ~160/147 = 146.305


Mapping generators: ~1323/1280, ~6615/5632
[[Support]]ing [[ET]]s: {{EDOs|189b, 525, 861, 1050, 1386, 1911, 2436}}


Optimal tuning (CTE): ~6615/5632 = 278.8998
=== 23-limit ===


{{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }}
Subgroup: 2.3.5.7.11.13.17.19.21.23


Badness: 0.0949
Comma list: 2500/2499, 3025/3024, 3060/3059, 3520/3519, 4096/4095, 6175/6174, 79135/79092


==== 13-limit ====
{{Mapping|legend=1| 21 0 59 82 24 111 114 38 95 | 0 13 -4 -9 19 -13 -11 20 0}}
Subgroup: 2.3.5.7.11.13


Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683
: Mapping generators: ~216/209 = 1\21, ~160/147


Mapping: [{{val| 21 4 39 98 58 107 }}, {{val| 0 6 2 -8 3 -6 }}]
[[Optimal tuning]] ([[CTE]]): ~160/147 = 146.308{{C}}


Optimal tuning (CTE): ~168/143 = 278.9058
[[Support]]ing [[ET]]s: {{EDOs|525, 861h, 1050f, 1911}}


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
== Blackmagic ==
Blackmagic is the 63 & 84 temperament, merging two systems which cover many large primes. It was named by [[User:Overthink|Overthink]] in 2026 as a twist on "blackjack" (which itself already refers to the 21-note [[MOS scale|mos]] of [[miracle]]), as well as because of its higher-limit properties. {{Todo|review}}


Badness: 0.0551
Subgroup: 2.3.5.7


==== 17-limit ====
Comma list: [[225/224]], {{Monzo|27 1 1 -11}}
Subgroup: 2.3.5.7.11.13.17


Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625
{{Mapping|legend=1| 21 0 82 59 | 0 1 -1 0 }}


Mapping: [{{val| 21 4 39 98 58 107 120 }}, {{val| 0 6 2 -8 3 -6 -7 }}]
: Mapping generators: ~16807/16384 = 1\21, ~3


Optimal tuning (CTE): ~168/143 = 278.9036
[[Optimal tuning]] ([[CWE]]): ~3/2 = 701.120{{C}}


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=1|21, 63, 84, 147}}


Badness: 0.0319
[[Badness]] (Sintel): 5.605


==== 19-limit ====
=== 2.3.5.7.11.13.23.29.31.43 subgroup ===
Subgroup: 2.3.5.7.11.13.17.19
Primes 17 and 19 could be included by mapping them to -1 and 1 generators respectively, though in practice this mapping only works in [[84edo]].


Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095
Subgroup: 2.3.5.7.11.13.23.29.31.43


Mapping: [{{val| 21 4 39 98 58 107 120 16 }}, {{val| 0 6 2 -8 3 -6 -7 15 }}]
Comma list: 155/154, 225/224, [[232/231]], [[300/299]], [[364/363]], 560/559, [[640/637]], [[1716/1715]]


Optimal tuning (CTE): ~168/143 = 278.9036
{{Mapping|legend=1| 21 0 82 59 106 111 95 102 104 114 | 0 1 -1 0 -1 -1 0 0 0 0 }}


{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
: Mapping generators: ~16807/16384 = 1\21, ~3


Badness: 0.0270
Optimal tuning ([[CWE]]): ~3/2 = 701.742{{C}}


==== 23-limit ====
{{Optimal ET sequence|legend=0|21, 63, 84, 147}}
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211
Badness (Sintel): 1.317


Mapping: [{{val| 21 34 49 58 73 77 85 91 95 }}, {{val| 0 -6 -2 8 -3 6 7 -15 0 }}]
{{Navbox fractional-octave}}
 
Optimal tuning (CTE): ~168/143 = 278.8971
 
{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
 
Badness: 0.0199
 
[[Category:21edo]]
[[Category:Temperament collections]]
[[Category:Rank 2]]
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